jossx“I'd like to know what grammar rule governs the following underlined words”
It's not really a grammar rule. These are participial constructions. They can be paraphrased in various ways.
College teachers don't go to high school,
HOPING to enroll [with the hope of enrolling] students which others have ignored. (This one should not have the comma.)
In the game, those kind of enemies make air units attack each other
, CREATING chaos [and by doing so, they create chaos] among enemy ranks.
Alan revealed the tunnel position, which could be seen from one hundred meters away, ENDANGERING [and by doing so, they endanger] the other prisoners.
1) No. These are not relative clauses, which are adjectival. They are closer in function to adverbial clauses.
2) No. The writer was not necessarily avoiding the repetition of a relative clause.
CJ